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I'm brushing up on my complex for an upcoming qual, and one of the questions had me use an alternate form of the residue theorem:

$$\oint_{\partial \Omega} \frac{f(\zeta)}{z-\zeta} d \zeta = 2 \pi i \sum_{j=1}^{n}\text{Res}(f,z_j)$$

for $f$ meromorphic on a bounded, star-shaped domain $D$ with boundary $\Gamma$, and $z \in \Omega$. I'm pretty sure it involves that the index of any path is one, but I can't figure out how to put it together.

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You have several things which are a bit confusing. You have a domain $D$ and also $\Omega$ in which you defined $D$ but not $\Omega$. In contrast, you used $\Omega$ in the question but not $D$. Am I correct in assuming that your $D$ and $\Omega$ are the same? –  EuYu Nov 11 '12 at 18:53

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